Call a fraction a "unit fraction" if it can be written as 1/n, where n is a positive integer.

How many more ways can the unit fraction 1/n be written as a sum of two (possibly equivalent) unit fractions than as a difference of two unit fractions?

let 1/n = 1/a + 1/b. So a = nb/(b-n) = n + n^2/(b-n). If F(n) represents the number of factors of n^2, then

# of sums = (F(n) + 1)/2

Similarly, 1/n = 1/a - 1/b implies a=nb/(b+n) = n - n^2/(b+n). Since denominators will have to be greater than n,

# of differences = (F(n) - 1)/2

# of sums - # of differences = 1

Posted by Dennis on 2006-10-13 11:47:11